Sorry for the delay. I missed that you had replied.
I noticed that if, say, I'm in a non-Private tab... and I press the tabs button, then I can only see the other non-Private tabs. So then I lose track of the other Private tabs I have open. This is frustrating.
I have lost track of the non-private tabs or Private tabs as well. I have iCloud Tabs set up on my MacBook Pro so I tend to notice the non-private tabs that way. I agree this is somewhat frustrating but assume it has to do with keeping the private tab websites and non-private websites separate with all the cross site tracking technologies out there. macOS Safari gets windows and tabs but iOS Safari only gets Tabs. I think the private button is more of a window switcher than anything else but that is just a guess.
Also, if I intentionally create a Private tab, then during my browsing I one a link into a new tab, the new tab doesn't inherit Privateness. Makes it difficult to control.
It doesn't work like Safari on my Mac.
So, this doesn't sound like you are doing anything wrong but also doesn't sound like it works as it should be.
I am curious about a setting I have on that may make a difference. If you go to you Settings App > Safari > in the TABS section, I have Open Links set to open In New Tab. I believe the default is In Background and wonder if that would make a difference somehow.
I tested and if I hold down on a link (from a Private tab) then select Open in New Tab, the new tab is also a Private Tab. You may be thinking these are non-private just because they do not say "- Private" in the address bar. The color is actually how you can tell. I have Dark Mode set so for me a non-private tab's address bar is grey but a private tab's address bar is black. I believe Light Mode would be white for non-private and dark grey for private.
Turn Private Browsing on or off on your iPhone or iPod touch - Apple Support
"While Private Browsing is on, Safari appears black or dark instead of white or gray." This is badly worded in my opinion. I have pictures below to show the difference on my iPhone.
Not Private:
Private:
If yours is truly opening a non-private tab from a private tab you may want to submit a bug report on Apple's feedback website and contact Apple Support to see if they can help. https://www.apple.com/feedback/
Again, sorry for the delay. I hope this clears it up a bit.
